In simple terms
A friendly intro before the formal notes — no formulas yet.
The Water Park of Circuits
A circuit moves charge the way a water park moves water. A pump lifts water and gives it energy; pipes and slides carry it and slow it down. Swap 'water' for 'charge' and you have a working mental model of every DC circuit.
A pump (the battery) raises water to a height, giving each litre energy — that energy per unit of charge is the electromotive force. The litres passing a point each second are the current. Slides and narrow pipes (resistors) oppose the flow, and a long thin pipe opposes it more than a short fat one — exactly what resistivity describes. The height the water actually falls through in the external pipes is the terminal potential difference, and it is a little less than the full pump height because the pump itself wastes some energy internally (internal resistance).
- 1
Start from the two things you can measure directly: current (flow of charge) and potential difference (energy given up per unit charge).
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Link them through resistance with Ohm's law, , for any component at a fixed temperature.
- 3
Combine components: resistances add in series, but in parallel the extra paths make the total resistance fall below the smallest branch.
- 4
Account for the real source: subtract the 'lost volts' inside the battery to get the terminal potential difference actually delivered to the circuit.
Explore the concept
Use the live diagram, PhET or GeoGebra sim, and synced steps — play it, drag controls, or tap a step.
Step-synced diagram — highlights what to look for in the simulation above.
Step 1
Start from the two things you can measure directly: current (flow of charge) and potential difference (energy given up per unit charge).
48 more simulations for this topic — run them in the Simulations section below
Simulations
Every simulation here runs the real model — try the steps on a card, then check what you see against the notes.
48 simulations · 5 to start with
Start herein this order — each one shows a different piece of the topic
- PhETStart here · 19702 9.3 · IB B.5
Resistance in a Wire
Change a wire’s resistivity, length and cross-sectional area; see the resistance update.
Why this one: Double the length and resistance doubles; double the area and it halves: R = rho-L/A.
Try this
- Double the length L — R doubles.
- Double the area A — R halves.
- Double ρ and halve L together — does R change?
Look for R = ρL/A: long, thin, high-resistivity wires resist most.
Simulation by PhET Interactive Simulations, University of Colorado Boulder · Licensed to MarkScheme (public licence CC BY-NC 4.0 since 2026-03-30)
- PhysicsHubStart here · 29702 10.2 · 9702 10.1 · 9702 9.2
Kirchhoff's Circuit Laws
Four DC circuits; set each EMF, internal resistance and the three resistances, drag a voltmeter and ammeter on; watch the KCL and KVL sums balance to zero
Why this one: Add internal resistance and watch terminal p.d. drop below the emf as current rises; loop sums stay zero.
Try this
- Pick the series circuit and read the current with the ammeter.
- Switch to parallel and compare the currents in each branch.
- Raise the internal resistance and read the terminal voltage.
Look for Currents into each junction sum to zero and the voltages around each loop sum to the EMF.
PhysicsHub (@mattqdev) · MIT
- oPhysicsStart here · 39702 10.1 · 9702 10.2 · IB B.5
Electric Circuit with Four Identical Lightbulbs
Combination circuit with four identical bulbs and three switches; predict then test brightness and current
Why this one: Predict each bulb's brightness in the series-parallel mix, then flip the switches to test it.
Try this
- Close one switch at a time and compare bulb brightness.
- Close all three switches.
- Open the switches in a different order.
Look for Bulbs in series share the current and glow dimmer; a parallel branch draws extra current from the source.
Simulation by Tom Walsh, oPhysics.com — made with GeoGebra · Licensed to MarkScheme (site: free for non-profit educational use; applets made with GeoGebra)
- The Physics ClassroomStart here · 49702 10.2 · IB B.5
Equivalent Resistance
Given a series, parallel or combination circuit, choose resistor values that produce a target equivalent resistance
Why this one: Choose resistor values that hit the target equivalent resistance in series, parallel and combination.
Try this
- Solve a series target.
- Solve a parallel target.
- Try a combination circuit.
Look for Series resistances add, and parallel resistances combine by reciprocals.
Physics Interactives by The Physics Classroom · Licensed to MarkScheme (site terms otherwise permit linking only)
- 3JCN PhysicsStart here · 59702 9.2 · 9702 9.3 · IB B.5
Ohm's Law
Vary voltage and resistance; read current and plot V-I
Why this one: Vary V and plot the V-I line; a constant gradient is what 'ohmic' means.
Try this
- Fix the resistance and double the voltage; read the current.
- Plot V against I and check for a straight line.
- Double the resistance and compare the gradient.
Look for For a fixed resistor V-I is a straight line through the origin with gradient R.
3JCN Physics Simulation by Thomas Nguyen · CC BY 4.0
More simulations43 more on this topic — core ones first
- PhETCore9702 10.1 · IB B.5
Circuit Construction Kit: DC — Virtual Lab
A real-looking bench: build a circuit, then measure with a non-ideal battery and meters.
Try this
- Build a battery and bulb; set the battery’s internal resistance (tap it) to 2 Ω.
- Measure the terminal p.d. with the voltmeter, then add bulbs in parallel — it drops.
- Compare the open-circuit reading with the loaded one — the difference is the “lost volts”.
Look for Terminal p.d. = E − Ir: more current drawn → bigger drop across internal resistance.
Simulation by PhET Interactive Simulations, University of Colorado Boulder · Licensed to MarkScheme (public licence CC BY-NC 4.0 since 2026-03-30)
- PhETCoreJava · best on a laptop9702 9.1 · 9.3 · IB B.5
Battery-Resistor Circuit
Vary the battery voltage and the resistance; watch electrons move through the resistor and the resistor heat up.
Try this
- Raise the voltage — electrons move faster and the ammeter reads higher.
- Raise the resistance — the current falls and the atoms in the resistor jiggle more.
- Set the voltage to zero — electrons still move, but with no net drift.
Look for Current is the net drift of electrons; collisions with lattice atoms transfer energy and heat the resistor.
Simulation by PhET Interactive Simulations, University of Colorado Boulder · Licensed to MarkScheme (public licence CC BY-NC 4.0 since 2026-03-30)
- PhETCoreJava · best on a laptop9702 9.1 · IB B.5
Signal Circuit
Close a switch on a long loop of wire and watch every electron start moving at once while each one drifts slowly.
Try this
- Close the switch — the bulb lights immediately.
- Watch one electron — it creeps along even though the signal was instant.
- Open the switch — all the electrons stop together.
Look for The electric field is set up almost instantly; drift velocity is tiny, so current starts everywhere at once.
Simulation by PhET Interactive Simulations, University of Colorado Boulder · Licensed to MarkScheme (public licence CC BY-NC 4.0 since 2026-03-30)
- The Physics ClassroomCore9702 9.1 · 9702 9.2 · 9702 10.1
DC Circuit Builder
A virtual circuit board: add resistors, bulbs, wires and ammeters, use a voltmeter, and build series, parallel and combination circuits
Try this
- Build a series circuit and read the ammeter.
- Rebuild the same resistors in parallel and compare.
- Measure the voltage across each resistor with the voltmeter.
Look for Current is the same everywhere in series, and voltage is the same across parallel branches.
Physics Interactives by The Physics Classroom · Licensed to MarkScheme (site terms otherwise permit linking only)
- SimuPhysicsCore9702 9.1 · IB B.5
Charge Carriers in Different Materials
A metal, an electrolyte, a semiconductor and an ionised gas side by side, each with its own carriers moving inside
Try this
- Compare the carriers in the metal and the electrolyte.
- Compare the semiconductor and the ionised gas.
- Compare the direction of conventional current in all four.
Look for Conventional current points the same way in all four whatever carries the charge.
Open on SimuPhysicsRuns on their siteSimuPhysics by Mohamed Abdelsalam · Licensed to MarkScheme (site publishes no licence; served with frame-ancestors self)
- SimuPhysicsCore9702 9.1 · 9702 11.2 · IB B.5
Charge is Quantised — Q = Ne
Dial in a charge and watch it divided by the elementary charge; a whole number shows that many electrons, a fraction shows what cannot exist
Try this
- Dial in a charge that gives a whole number.
- Dial in one that gives a fraction.
- Find the smallest charge that works.
Look for Any real charge is a whole-number multiple of e.
Open on SimuPhysicsRuns on their siteSimuPhysics by Mohamed Abdelsalam · Licensed to MarkScheme (site publishes no licence; served with frame-ancestors self)
Key formulas
Tap any symbol to reveal exactly what it means and its units.
Tap a symbol — great for exam definitions
Full topic notes
Formal explanation with the rigour you need for the exam.
Charge, current and potential difference
Electric current is the rate at which charge flows past a point. If a charge passes in a time , the current is , measured in amperes (1 A = 1 C s⁻¹). By convention we take current as the direction positive charge would move, even though in a metal it is electrons that drift the other way. Potential difference is the partner idea: it measures energy, not flow. The p.d. across a component is the electrical energy converted to other forms per unit charge passing through it, , measured in volts (1 V = 1 J C⁻¹).
Current is a flow of charge; potential difference is energy transferred per unit charge.
Electromotive force is energy SUPPLIED per unit charge by a source; p.d. is energy USED per unit charge in a component. Both are in volts.
Conventional current flows from + to − in the external circuit; electron flow is opposite.
Charge is conserved: it is neither created nor destroyed as it moves round a circuit.
Resistance, Ohm's law and resistivity
Resistance measures how strongly a component opposes current, defined as and measured in ohms (). For many conductors at constant temperature the current is proportional to the p.d.: this is Ohm's law, written with constant. Resistance also depends on the physical shape and material of a conductor. A longer wire has more resistance and a thicker wire has less, captured by the resistivity relation , where is the resistivity of the material (unit m), its length and its cross-sectional area.
Ohm's law applies only to ohmic conductors at constant temperature — it is not universal.
Resistance length and : doubling the length doubles ; doubling the cross-sectional area halves .
Resistivity is a property of the MATERIAL; resistance is a property of a particular SAMPLE of it.
Rearrange Ohm's law freely: and .
Ohmic and non-ohmic behaviour: I–V characteristics
An I–V characteristic is a graph of current against potential difference for a component. Its shape tells you at a glance whether the component obeys Ohm's law. A metal resistor at constant temperature gives a straight line through the origin — its resistance () is constant, so it is ohmic. A filament lamp gives an S-shaped curve that flattens as the voltage rises: heating the filament raises its resistance, so it is non-ohmic. A diode is strongly non-ohmic — it conducts almost nothing until a threshold forward voltage, then the current rises steeply, and it blocks current in the reverse direction. For any point on any of these graphs the resistance is still ; only for the straight ohmic line is that ratio constant.
Resistor (ohmic): straight line through the origin; constant resistance.
Filament lamp (non-ohmic): curve that flattens; resistance increases as it heats up.
Diode (non-ohmic): negligible current below the threshold voltage, then a steep rise; blocks reverse current.
Resistance at any point is always (the reciprocal gradient of a line from the origin), not the gradient of the curve itself.
Resistors in series and parallel
In series, components form a single path, so the SAME current flows through each and the source p.d. is shared between them; resistances simply add. In parallel, components connect across the same two points, so each has the SAME p.d. across it and the current splits between the branches; here the reciprocals of the resistances add. Because extra parallel paths make it easier for charge to flow, the combined parallel resistance is always smaller than the smallest single branch — a fact worth using as a check on every parallel calculation.
Electrical power
Power is the rate at which electrical energy is transferred. Combining (energy per charge times charge per second) with Ohm's law gives three equivalent forms. Use whichever version fits the two quantities you already know: when you have p.d. and current, when you have current and resistance, and when you have p.d. and resistance. All give power in watts (W).
EMF, internal resistance and terminal potential difference
A real cell is not a perfect source. It has an internal resistance in series with an ideal EMF . When a current flows, some energy is dissipated inside the cell, producing 'lost volts' equal to . The voltage actually available to the external circuit — the terminal potential difference — is therefore , which is also just the p.d. across the external load. For a single external resistor the EMF is shared between load and internal resistance, giving .
EMF is the total energy per unit charge supplied by the source.
'Lost volts' is the p.d. dropped across the internal resistance.
Terminal p.d. : always less than when current flows, and equal to only when (open circuit).
A graph of against is a straight line with intercept and gradient .
Kirchhoff's laws
Kirchhoff's two laws are the bookkeeping rules that let you analyse any circuit, however it is wired. The first law (the junction rule) says the total current flowing into any junction equals the total current flowing out — nothing more than conservation of charge, and the reason series current is constant while parallel current splits. The second law (the loop rule) says that around any closed loop the sum of the EMFs equals the sum of the potential-difference drops — a statement of conservation of energy, since a unit charge returning to its start must have gained exactly as much energy as it lost.
First law (junction / charge): at every junction.
Second law (loop / energy): around every closed loop.
The junction rule explains why current is the same all round a series loop and why branch currents add up in parallel.
The loop rule explains why the shared p.d.s across series resistors add up to the source's terminal p.d.
Common mistakes examiners penalise
Thinking current is 'used up' in series — current is identical everywhere in a single loop (conservation of charge). It is ENERGY that is transferred at each resistor, not current.
Forgetting to invert the parallel result — adding reciprocals gives , not . Always take the final reciprocal, and check the total is below the smallest branch.
Setting terminal p.d. equal to EMF — whenever current flows there are 'lost volts' , so . They are equal only on open circuit.
Treating a filament lamp or diode as ohmic — these are non-ohmic; their resistance changes, so you cannot use a single fixed and their I–V graphs are curved.
Confusing EMF with potential difference — EMF is energy supplied per unit charge by the source; p.d. is energy dissipated per unit charge in a component.
Mixing up the power formula — and are not interchangeable inputs: use the current with and the p.d. across THAT resistor with .
Confusing resistivity and resistance — resistivity is a material property (unit m); resistance depends on the sample's length and area through .
Model answer — marked the way our engine marks it
This is the showcase for a calculation topic. In Paper 2 the marks are analytic: each is tied to a specific piece of working — a method mark (M) or an answer mark (A) — and, crucially, the method marks and error-carried-forward (ECF) mean a wrong number early on does not cost you every mark that follows. That protection only exists if your method is written down. Study how each mark below is earned by a specific line.
Where this leads
The habits built here — track the current with Kirchhoff's junction rule, track the energy with the loop rule, and reach for the right form of the power equation — carry straight into the rest of electricity. Potential dividers are just series p.d.-sharing used deliberately; capacitor and RC work reuses charge, current and energy; and electromagnetic induction generates the very EMFs you have learned to handle here. Master the three-step circuit method — combine the resistances, find the current, then work out each p.d. — and the harder circuit questions become variations on a method you already own.
Worked examples
See the formulas applied — reveal one step at a time, like the exam.
A 12 V battery of negligible internal resistance is connected to a 4.0 Ω resistor in series with a parallel combination of a 12 Ω resistor and a 6.0 Ω resistor. Calculate (a) the total resistance of the circuit, (b) the current drawn from the battery, and (c) the current in the 6.0 Ω resistor. [6]
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Step 1 — combine the parallel pair. , so . [M1 method, A1 value]
A resistor of resistance 15 Ω carries a steady current of 0.40 A. (a) Calculate the potential difference across it. (b) Calculate the power dissipated, and confirm your answer by a second method. [4]
- 1
Step 1 — potential difference (Ohm's law). . [M1 method, A1 value]
A battery of EMF 9.0 V and internal resistance 1.5 Ω is connected to an external resistor of 6.0 Ω. Calculate (a) the current in the circuit, (b) the terminal potential difference, and (c) the power dissipated inside the battery. [5]
- 1
Step 1 — current from the loop equation. , so . [M1, A1]
A cell of EMF 6.0 V and internal resistance 0.50 Ω is connected to a 2.5 Ω resistor. Calculate the current in the circuit and the terminal potential difference. [3]
- 1
Model answer — full working.
How it all connects
The big idea sits in the middle — tap a linked idea to explore the link.
Tap a linked idea to see how it connects back to the main topic — that connection is what examiners reward.
Glossary
Key terms for this topic — skim now; the Check step will test them.
- Electric current ()
The rate of flow of electric charge: . Unit: ampere (A), where . Conventional current is the direction positive charge would move.
- Potential difference ()
The energy transferred from electrical to other forms per unit charge passing through a component: . Unit: volt (V), where .
- Electromotive force ()
The energy supplied per unit charge by a source (e.g. a cell), converting chemical to electrical energy. Also measured in volts. It is NOT a force despite the name.
- Resistance ()
The opposition to current, defined as . Unit: ohm (), where .
- Ohm's law
For a conductor at constant temperature, current is proportional to potential difference: with constant. It is a property of ohmic conductors, not a universal law.
- Resistivity ()
A material property: , so resistance rises with length and falls with cross-sectional area. Unit of : .
- Ohmic vs non-ohmic
Ohmic: constant , straight-line I–V graph through the origin (metal resistor). Non-ohmic: changes, so the I–V graph curves (filament lamp, diode).
- Resistors in series
Same current through each; the source p.d. is shared between them.
- Resistors in parallel
Same p.d. across each; the current splits between branches. The total is always LESS than the smallest branch.
- Electrical power
. Unit: watt (W). Choose the version that uses the two quantities you already know.
- Internal resistance ()
The resistance inside a source itself. When current flows it dissipates energy as heat inside the cell, producing 'lost volts' equal to .
- Terminal potential difference
The p.d. actually delivered across the source's terminals: . It equals the p.d. across the external load and is always less than when current flows.
- EMF and internal resistance
For a source driving a single external resistor: . The EMF is shared between the load () and the internal resistance ().
- Kirchhoff's first law (junction)
The sum of currents into a junction equals the sum out — a statement of conservation of CHARGE. This is why current splits in parallel and recombines.
- Kirchhoff's second law (loop)
Around any closed loop, the sum of EMFs equals the sum of the p.d. drops — a statement of conservation of ENERGY.
Name it
Read the meaning, then pick which of this lesson’s terms it describes. Miss one and you see what your choice really means.
The sum of currents into a junction equals the sum out — a statement of conservation of CHARGE. This is why current splits in parallel and recombines.
Quick check
Write your answer first, then compare it with the model one — the gap is what you would have lost.
Teach it back
If you can explain it simply, you own it — gaps here are marks you’d lose.
Teach it back
Explain this topic as if teaching a friend. We name the gaps an examiner would still dock.
Revision flashcards
Guess first, then flip — retrieval beats re-reading.
Key takeaways
Review these before you close the topic — retrieval beats re-reading.
Current is a flow of charge; potential difference is energy transferred per unit charge.
Electromotive force is energy SUPPLIED per unit charge by a source; p.d. is energy USED per unit charge in a component. Both are in volts.
Conventional current flows from + to − in the external circuit; electron flow is opposite.
Charge is conserved: it is neither created nor destroyed as it moves round a circuit.
Practice — then mark it
The whole point: a real Cambridge question, marked mark-by-mark.
Get a Paper 2 circuit calculation marked: find the current and terminal potential difference with full working
Get a Paper 2 circuit calculation marked: find the current and terminal potential difference with full working
Extra simulations & links
PhET, GeoGebra and other curated tools — open in a new tab.
Frequently asked
Checkpoint
One marked question is worth ten re-reads — close the loop before you move on.
Reading it isn’t knowing it — prove it.
Before you move on: do Get a Paper 2 circuit calculation marked: find the current and terminal potential difference with full working on paper, snap a photo, and get examiner-style feedback on exactly where you win and lose marks.
Discuss Current and circuits
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